
// Make newform 6384.2.a.cc in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6384_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6384_2_a_cc();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_6384_2_a_cc();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [6, 16, 2, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 2, 0, 0, 0], [0, -4, -1, 1, 0], [-6, -2, 2, 0, 0], [11, 12, -10, -4, 2]];
    Rf_basisdens := [1, 1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_6384_a();"
function MakeCharacter_6384_a()
    N := 6384;
    order := 1;
    char_gens := [799, 4789, 2129, 913, 1009];
    v := [1, 1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_6384_a_Hecke(Kf)
    return MakeCharacter_6384_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [0, 0, 1, 0, 0], [-1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [2, 0, 0, -1, 0], [-1, 0, -1, 0, -1], [-1, 0, 0, 0, 0], [0, -1, 0, 0, 0], [0, 1, 1, 0, 0], [-2, 1, 0, 0, 0], [1, 0, 2, 0, -1], [1, 0, 0, -1, 1], [-3, -1, 2, 0, -1], [6, -1, 1, 0, 0], [1, 1, -1, -2, 1], [0, 2, 0, 0, 0], [3, 0, 2, 0, 1], [-1, 1, 2, -1, 1], [-1, -1, 1, -1, -1], [1, 0, 0, 2, -1], [-2, 1, 2, -1, 0], [5, 1, -1, 2, -1], [1, 0, -4, -1, 1], [-5, 3, 0, 1, -1], [1, 2, 1, 0, 1], [2, 0, 0, -2, -2], [0, -3, 1, -1, 0], [2, -2, 2, 0, 0], [-2, -1, 1, 2, -2], [-1, -1, -2, 1, 1], [7, 1, 1, 0, 1], [4, -2, 0, -2, -2], [3, 0, 0, 2, -1], [6, 2, -2, 0, 0], [-2, -1, 0, -3, 0], [3, 0, -2, 2, 1], [-1, 1, 2, -2, 1], [12, 0, 2, -2, 0], [1, -2, 0, -1, 1], [2, -1, -1, 1, 2], [7, 1, 4, -1, -1], [0, 1, 4, 2, 0], [1, 2, 2, 0, -1], [4, -2, 0, -4, -2], [4, -2, -2, 0, 0], [-3, 3, -4, 1, -1], [12, 1, 4, 0, 2], [2, -2, 2, 0, 2], [0, 0, -2, 2, -2], [4, 0, 2, 4, -2], [-2, -3, 2, -2, -2], [7, -5, 0, 1, -1], [4, 1, 1, -2, -2], [13, 0, -2, -1, 1], [-2, 1, -2, 0, -2], [-3, 0, 2, 1, -3], [7, 2, 2, -2, 3], [-2, 3, 2, 2, 2], [13, 1, -3, 0, 1], [2, -4, 2, 0, -2], [-11, 6, 0, 1, 1], [-2, 1, 0, -4, 0], [7, 1, -3, -2, 1], [1, 2, -2, 0, -1], [0, -1, 1, -4, 0], [8, -1, -4, 3, -2], [6, 4, -2, -2, 4], [0, 3, -2, 4, 0], [-8, 0, 4, -2, 2], [-8, -2, -1, 2, 4], [14, 3, 0, 0, 2], [16, 0, 2, 0, 0], [2, -6, 0, 2, 0], [-2, -3, 0, 3, 0], [12, 0, 2, -2, 0], [-4, -4, 2, -2, 2], [2, -2, 4, -2, 0], [13, 1, 3, 2, 1], [0, -2, -4, -3, -2], [20, -1, 3, 2, 2], [-13, -2, 0, 2, -3], [-4, -1, -7, 1, -4], [7, -3, 4, -3, -1], [-4, -1, -4, 2, -2], [24, -3, 6, 0, 0], [11, -1, 3, -2, -1], [8, -1, 2, -4, 0], [-5, 2, 1, 0, -5], [-10, 0, 0, 0, -2], [3, -1, -3, 0, -3], [8, 3, 7, -2, 2], [7, -1, 4, -5, -3], [0, 7, -2, 0, 0], [6, -2, 0, 4, 2], [17, -3, -3, -4, -1], [-1, 0, 2, 1, -1], [1, -2, 2, 1, 1], [-12, 1, 0, 6, 2], [-2, 4, 0, 4, -4], [-12, 1, -2, -5, 2], [-16, -4, -2, 2, 2], [-6, 0, -2, -2, -6], [23, -1, 9, -2, -1], [-4, 0, -4, -2, 4], [1, -2, 6, 4, -1], [14, -1, 1, 0, -4], [10, -2, 9, 4, -2], [-10, 1, -5, 1, 2], [3, -1, -4, 3, -5], [-8, 2, 0, 0, 0], [-2, 3, -6, -6, 2], [-10, 0, -4, 2, -4], [7, 4, 8, -2, 3], [3, -1, -6, -6, 5], [22, 1, 3, 4, -2], [-20, 0, -6, 4, 0], [14, 1, 5, 2, -4], [8, 0, -4, 0, 2], [-11, -1, 3, -7, 1], [18, 4, -4, -5, 4], [-15, 0, -6, -2, -5], [1, 0, 2, -5, 1], [-10, 1, 7, -1, 2], [-9, 6, -6, 2, -1], [10, 2, -4, 2, 4], [0, 3, -2, -4, 4], [3, -5, 3, -8, -3], [15, 0, 2, 2, -1], [-3, 2, 4, 0, -5], [7, 3, -6, 4, -3], [-10, 1, -3, -1, 2], [-16, 5, -4, 3, -2], [14, -2, -4, -2, 0], [22, -2, 5, -2, -2], [7, 2, -6, -4, 5], [-3, -2, -4, 3, -3], [-4, 3, 0, 2, -2], [-9, 4, -8, -5, 3], [14, 2, 8, -2, -4], [-14, -2, 0, -4, 6], [20, -2, 2, -2, 2], [-13, 1, -2, 8, 1], [1, 5, 5, -3, 1], [-1, -3, 4, 3, 3], [10, 2, -4, -4, -2], [-26, -2, 0, 0, -4], [9, -2, -4, -9, -3], [-10, 2, 4, 0, -2], [-16, -1, 7, 1, -4], [-10, 4, 8, 4, 0], [10, 2, 7, 2, -2], [-22, -2, -4, 4, -2], [14, 6, -2, 4, 2], [-3, 3, 4, 3, -1], [-11, 5, -5, -5, 1], [-4, 4, -8, 0, 4], [9, -4, 7, 6, -3], [-10, -2, 0, -2, 0], [-13, 2, -2, -7, -1], [0, 5, -4, 6, 0], [-5, -1, -9, -2, 7], [-11, 1, -10, 6, 3], [-8, -6, -4, -2, 4], [15, 5, -5, 2, 3], [-8, 0, -12, -2, 4], [-5, 1, -4, -3, -3], [-1, 0, 6, -2, -3], [-3, -2, 8, 4, -5], [-3, 2, -3, -4, -3], [-12, -1, -3, 1, 0], [16, 0, 0, -1, 2], [-4, -5, -2, 2, -4], [14, 7, -2, -2, 6], [-8, 2, -4, -8, 0], [3, 0, -3, -2, 3], [-5, 1, 2, -4, -3], [-10, -6, -4, 4, -4], [8, -5, 12, -2, -2], [-6, 4, -8, -4, 4], [-32, 6, 8, 8, 0], [-24, 6, -12, -4, 0], [-7, 6, -4, -4, 3], [9, 0, 4, 3, -3], [-36, 1, 0, -2, 0], [23, -2, -3, -4, -1], [-22, 6, 2, 2, 0], [-12, 1, -8, 4, 2], [1, -2, -12, -6, -1], [2, 1, -9, 6, -4], [-20, -2, 0, -1, 2], [-4, -3, 4, 2, -4], [-4, 1, -8, 5, -2], [-19, 2, 2, 1, 5], [2, 7, 0, -8, 2], [-7, -4, -14, 3, -7], [4, -5, 6, 2, -4], [-10, 2, -8, -6, 0], [4, 2, 11, -4, 4], [-16, -1, 3, 6, 2], [-21, 1, -13, 2, 3], [-20, -6, 4, -2, 0], [-15, 3, -16, 1, 1], [24, 2, 0, -1, -2], [21, -1, -11, -6, 3], [-12, 4, 1, -6, 0], [26, -10, 2, -8, -2], [9, -1, -7, -1, -3], [-16, 6, 6, 4, 2], [8, -5, 4, -8, -2], [6, -5, -10, 0, -2], [-2, -4, -3, 0, -2], [-25, 5, -10, -6, 1], [-4, -6, -2, 8, -4], [-34, -2, -4, 4, 2], [-2, -2, -4, -7, 8], [7, -5, 2, -10, -3], [17, 2, -2, -1, 5], [-27, -3, -1, 5, -3], [-20, -4, 6, 0, -2], [-4, -8, -2, -4, 6], [-20, -6, 4, -6, 0], [6, -7, -3, 0, -6], [4, 8, -8, 2, 0], [-12, 4, -8, -4, 0], [-34, -6, -4, 2, 0], [-3, -2, 10, 3, -3], [-33, 7, 11, 3, 3], [12, 3, 12, -4, 2], [28, 5, 10, -2, 4], [-20, 4, -4, -4, 2], [-2, 5, -12, 4, 0], [-15, -5, -4, -1, -1], [35, 3, 9, 0, -1], [-2, 4, 0, 0, -6], [11, -3, -1, -2, -7], [5, 0, 8, -6, 3], [-6, 6, 12, -2, 0], [22, -9, -2, -2, -2], [14, 0, 6, -6, 2], [-17, -3, -1, 5, -5], [-11, -2, 2, -2, -3], [-2, 2, 6, 4, -6], [-35, 6, -6, 2, -1], [-5, -6, 2, 3, -5], [-25, 7, 3, -2, 5], [11, 6, -4, -2, 7], [28, 2, 2, 4, 4], [-8, -1, 2, -3, -6], [-10, 7, 4, 8, 2], [0, -10, 0, -2, -2], [5, 4, 1, 6, 5], [-32, -1, 2, -10, 4], [-18, 2, -4, -1, 4], [-12, -8, -1, 4, 0], [16, 3, -12, -4, 4], [-2, 2, 8, -2, -4], [6, 5, 12, 2, 8], [-29, 2, 2, 3, 7], [1, 6, 2, -6, -1], [-48, 5, 6, 1, 2], [24, -1, 9, 7, 0], [21, -6, -4, 0, -5], [40, 8, 0, -3, 2], [-34, 2, 1, 0, 2], [-40, 5, -12, -8, -2], [-20, 11, -5, 2, 4], [3, 8, 5, -2, 3], [-26, 7, -2, 3, 0], [-6, -4, 2, 10, 0], [17, 3, 0, -1, -3], [17, -5, -4, 3, -5], [6, 4, 6, 2, -8], [-18, 2, -4, -10, 2], [-7, 3, 12, 9, -3], [29, -6, 12, 2, 1], [36, -11, -2, -6, 0], [-3, 4, 2, 2, -5], [13, 0, -2, 8, -1], [-4, 3, -7, -4, 2], [-16, -4, -6, -2, -4], [-7, 5, -10, 0, 7], [-11, 5, 5, 8, -5], [-15, 3, 4, 7, 1], [29, -12, 10, 2, 1], [14, 3, -10, -8, 2], [36, -8, 0, 0, -2], [13, 3, -3, 6, 1], [-11, -12, -12, 6, -3], [-17, -12, -2, 11, -5], [42, 0, 2, -4, 0], [20, -3, -1, -6, 2], [-15, 1, 9, -4, -3], [4, -2, -4, 0, 4], [-20, -4, 12, 6, -10], [5, 6, -9, 6, 5], [22, -9, 4, 6, 0], [-2, 0, -3, 0, -2], [8, 4, 2, 2, 8], [11, -2, -4, 6, -5], [29, -6, 2, 2, 3], [-2, 1, 9, 12, -2], [-42, -2, 4, -2, 2], [-43, -1, 5, 3, -7], [17, -11, -14, 2, -5], [-4, -2, 2, -10, 6], [17, -5, 1, -3, -7], [20, 0, 12, -5, -6], [12, -7, -2, 0, 0], [30, -7, 2, 4, -2], [-17, -5, -7, 2, 1], [-30, 10, -10, -4, 0], [-24, -3, -13, 4, 0], [-26, 5, 4, 5, 0], [28, 1, 8, -6, 4], [3, 3, 0, 5, 7], [-18, 3, 1, -6, 0], [-12, 7, -7, -5, 0], [-29, 2, 0, -8, 5], [14, -2, 0, -4, -4], [-4, 8, 8, 6, 4], [-2, 1, 6, 4, 2], [17, -10, -14, 1, -7], [2, -1, -4, -3, 4], [-16, 4, 14, -2, 2], [33, -1, 20, 1, 1], [4, -3, -2, 9, -6], [10, 2, -4, -12, 6], [22, -1, 1, 1, -2], [-4, 2, -4, -4, 2], [-16, 6, 0, 2, 6], [19, 5, 9, -2, 7], [-21, 14, -4, 6, -1], [27, -1, 19, -10, -3], [22, -2, 4, 8, -6], [27, -13, -3, -4, -3], [-2, 0, 8, -4, 8], [-13, -2, 4, 5, 11], [6, 10, -14, 4, 4], [23, -6, -2, 10, -5], [-22, -2, -4, 2, 0], [-2, -8, 12, 4, -8], [7, 0, 8, -9, 7], [23, 1, 4, 3, 1], [10, -5, 5, -2, -2], [-4, 3, -1, 11, 8], [-4, -2, -2, 0, 6], [-3, -1, -8, 1, -1], [4, -2, -4, 0, -8], [47, 2, -3, 4, -1], [-14, 4, -4, 2, -6], [-25, 6, -8, 0, -3], [12, 4, 0, -10, 6], [-16, 2, -16, 2, 4], [18, 0, -12, -2, 4], [-3, -4, 6, -5, -3], [-30, -1, 1, 10, 0], [43, -5, -1, 0, 1], [10, -4, -14, 8, -8], [-25, -9, 5, -1, -1], [24, 6, 0, -2, -2], [-10, 6, 5, 2, -10], [-7, 3, -7, 3, -7], [11, -1, 13, 7, -1], [19, 0, -4, 2, -9], [-57, -4, 2, 2, 1], [-39, -2, 7, 0, -3], [19, -13, 18, -3, -3], [-7, 2, -14, 0, -9], [-8, 9, -20, 6, 4], [4, -13, -4, 6, -6], [-36, 3, -2, -8, 0], [17, -9, -1, 0, 5], [29, -6, -10, -14, 1], [4, 6, 8, 6, 2], [0, -4, 8, -2, 0], [26, 1, -8, -10, 2], [4, 8, 16, -4, 2], [6, 2, -15, 8, -6], [-13, -7, 0, -1, 3], [13, -4, -6, 5, 1], [-24, -6, 0, 4, -6], [31, -9, -2, -4, -3], [-20, -8, 0, 2, -2], [-4, 9, 4, 0, 10], [0, -8, -2, 4, 0], [-5, 7, 10, 8, 9], [28, -8, -14, -2, 2], [-47, -3, -6, 2, 3], [-22, 0, -22, 0, 2], [-15, 2, -4, 2, 7], [7, 10, -8, 1, 7], [-4, 15, 4, 4, -4], [-40, 4, 4, 8, 4], [34, -11, 5, -1, -2], [6, -12, -2, 2, 4], [6, -8, 0, 6, 2], [-9, 6, -12, -1, -9], [-5, -3, 2, -7, -3], [4, -2, -6, 6, -2], [18, -4, 4, 1, 4], [16, 10, -8, 2, -2], [-4, 1, 10, 3, -2], [-19, 2, 2, 3, 5], [23, -2, 2, -5, -1], [46, -10, -6, -8, -6], [-14, -14, 8, 2, -4], [-12, -5, 7, -8, 12], [-4, 6, -4, -2, 4], [-20, 9, 11, 0, -2], [5, 13, -8, 1, -3], [10, -4, -2, 8, 4], [-11, 5, 1, 15, 1], [23, 1, 6, -4, -11], [34, 0, -16, 8, -4], [-22, -4, 3, -14, 2], [-23, 13, -7, -1, 1], [-20, 11, -12, 2, -6], [12, 2, 7, 10, 0], [30, 12, 12, -8, 6], [-57, 1, -3, -10, 3], [15, -9, -12, -5, -5], [-4, 3, -18, -4, 8], [-6, -2, -4, 8, -6], [-42, 2, 4, 7, -8], [-13, 2, 4, 3, -13], [28, 7, 3, -3, 4], [-2, -6, -10, -2, -8], [9, -2, 8, 2, -7], [-18, -1, -11, 4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6384_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6384_2_a_cc();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_6384_2_a_cc(:prec:=5)
    chi := MakeCharacter_6384_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_6384_2_a_cc();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_6384_2_a_cc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6384_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-48, 68, -8, -16, 2, 1]>,<11,R![192, 256, 16, -32, -2, 1]>,<13,R![-256, 32, 112, -8, -8, 1]>,<17,R![3168, 1108, -176, -72, 2, 1]>,<23,R![-192, 256, -16, -32, 2, 1]>],Snew);
    return Vf;
end function;